Quantum chaos challenges many-body localization

Abstract

Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g = log 10 (tH/t Th ), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t Th and the Heisenberg time tH, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t Th ≈ tH, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.

Signatures of a critical point in the many-body localization transition

“…Both the position and the value of this maximum are shown to increase linearly with the system size, at least for systems small enough to be exactly diagonalized. This finite-size scaling is compatible with recent results suggesting that the MBL transition belongs to the Berezinskii-Kosterlitz-Thouless class [47]. We also show that the maximum of the kurtosis excess happens at the same disorder strength at which spectral statistics cease to follow the universal random matrix theory behavior at any scale -when Thouless and Heisenberg energies coincide.…”

“…These results are compatible with the values obtained in Refs. [28,47] from spectral statistics and entanglement entropy (they find ω 1 ≈ 0.25). It is worth to remark that this linear increase does not necessarily imply that ω c (L) → ∞ in the thermodynamic limit.…”

“…A variety of ergodicity breaking indicators [15,23,24,28,[36][37][38][39][40][41][42][43] have been used in the past to try and identify a hypothetical value of disorder strength that, in the thermodynamic limit, completely separates the ergodic and the nonergodic phases. The search for such a critical point implicitly assumes a nature of real phase transition in MBL systems, although this has not been proved and some apparent contradictions seem to exist which have led to much debate in the community [28,[44][45][46][47]. To obtain this value, different kinds of involved finite size scalings of such indicators have been employed.…”

“…To obtain this value, different kinds of involved finite size scalings of such indicators have been employed. However, the problem is not simple at all and a strong dependence on the particular indicator and the number of sites means that some of the previously obtained values for the critical point are often incompatible with each other; today much uncertainty remains about the properties and the phenomenology of the MBL transition [19,23,28,30,38,39,44,45,[47][48][49][50][51].…”

Route to Extend the Lifetime of a Discrete Time Crystal in a Finite Spin Chain without Disorder

“…T at long times. However, the requirement of MBL is very restrictive, thereby making the realization of a DTC in large systems extremely challenging [57,58]. Furthermore, MBL can lead to long-lived transient dynamics, thereby making it difficult to access the long-time behavior of the system in current experiments [59].…”